wajsberg_hoops - MathStructures

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=====Wajsberg hoops=====

====Definition====
A \emph{Wajsberg hoop} is a [[hoop]] $\mathbf{A}=\langle A, \cdot, \rightarrow, 1\rangle$ such that

$(x\rightarrow y)\rightarrow y = (y\rightarrow x)\rightarrow x$

Remark: Lattice operations are term-definable by $x\wedge y=x\cdot(x\rightarrow y)$ and $x\vee y=(x\rightarrow y)\rightarrow y$.

==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be Wajsberg hoops. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: 

$h(x\cdot y)=h(x)\cdot h(y)$, $h(x\rightarrow y)=h(x)\rightarrow h(y) $, $h(1)=1$

====Examples====
Example 1: 

====Basic results====


====Properties====
^[[Classtype]]  |variety |
^[[Equational theory]]  |decidable |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  | |
^[[Locally finite]]  |no |
^[[Residual size]]  | |
^[[Congruence distributive]]  |yes |
^[[Congruence modular]]  |yes |
^[[Congruence n-permutable]]  | |
Congruence regular & yes Radim Belohlovek, \emph{On the regularity of MV-algebras and Wajsberg hoops},
Algebra Universalis,
\textbf{44}, 2000, 375--377[[MRreview]]\\\hline
^[[Congruence uniform]]  | |
^[[Congruence extension property]]  | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  | |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
====Finite members====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &\\
f(4)= &\\
f(5)= &\\
f(6)= &\\
f(7)= &\\
\end{array}$

====Subclasses====
[[Generalized Boolean algebras]] 

[[MV-algebras]] 

====Superclasses====
[[Hoops]] 


====References====

[(Ln19xx>
)]

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